1. Introduction

Recent studies of observations (Feldstein and Lee 1998; Hartmann and Lo 1998), general circulation models (Feldstein and Lee 1996), and mechanistic dynamical models (Robinson 1991, 1996; Yu and Hartmann 1993;Lee and Feldstein 1996) appear to be converging upon a dynamical understanding of variations in the zonal index.¹ Specifically,

• anomalies in the zonally averaged zonal wind often persist for 10 or more days;

• such persistence is due, at least in part, to the reinforcement of anomalies in the zonally averaged zonal flow by transient eddy momentum fluxes; this is denoted positive eddy feedback; and

• only eddies with relatively high frequencies, periods of 10 or fewer days, contribute to the positive eddy feedback. Eddies with longer timescales either drive variations in the zonal index stochastically or, in some models, contribute systematically to their dissipation.

The dynamical basis of positive eddy feedback remains unclear. Hartmann (1995) and Hartmann and Zuercher (1998) argue that positive eddy feedback derives from the sensitive dependence of baroclinic life cycles on the preexisting meridional shear. While their life cycle calculations are convincing, it is not clear how to carry the results of initial-value life cycle experiments over to the real atmosphere where baroclinic eddies are always present. An alternative suggestion was made by Robinson (1996), who argued that the baroclinic component of the zonal index was crucial for positive eddy feedback. Robinson based his argument on the meridional coincidence, found in a mechanistic model, between anomalously strong poleward eddy heat fluxes and anomalously strong westerlies.

The proposed feedback loop is as follows: through the action of surface drag a region of anomalously strong barotropic westerlies becomes a region of enhanced baroclinicity. Baroclinic eddies are generated more vigorously in such a region, and as the eddies propagate meridionally away from their latitudes of generation, their momentum fluxes reinforce the anomalously strong barotropic westerlies, thereby closing the loop. It should be emphasized that this argument, and its implications for the relative timing of increases in heat and momentum fluxes, applies only to that portion of the variability in eddy activity that is organized by variations in the large-scale flow. Clearly, much of the variability in eddy activity is effectively stochastic. This is true even in mechanistic models (e.g., Robinson and Qin 1992).

Here we attempt to quantify under what conditions a baroclinic positive eddy feedback can work. The acceleration of barotropic westerlies in a region of baroclinic eddy generation over the course of a nonlinear baroclinic life cycle is an established result (e.g., Simmons and Hoskins 1978), but for this feedback loop to function, the zonally averaged low-level baroclinicity in this region must be reinforced in the response of the mean flow to eddy transports of heat and momentum. A key result of this paper is that this is indeed possible, but only through the action of surface drag.

We also attempt to explain some more subtle aspects of the zonal index that have emerged from observations and models:

1. the poleward drift of anomalies in the zonally averaged zonal wind (Feldstein 1998);

2. the meridionally banded structure of anomalies in both the zonal wind and its eddy forcing (Robinson 1996; Feldstein and Lee 1998);

3. in a mechanistic model, the dependence of positive eddy feedback on surface drag (Robinson 1996); and

4. in a mechanistic model (Robinson 1994, 1996), the frequency dependence of the positive eddy feedback.

The last two points need elucidation. While it is impossible to unequivocally demonstrate the existence of positive feedback in observations, and it is very difficult to do so in a full GCM, simple models can be manipulated in such a way as to verify the existence of and to quantify the strength of feedbacks operating within these models. Robinson (1994) did this by inserting an unphysical zonally symmetric source of zonal momentum into the equations of motion of a fully nonlinear, two-level, global, primitive equations model. The source of momentum was barotropic and was dipolar in latitude, thus mimicking the structure of zonal momentum variations generated internally by the model. The source oscillated in time, and experiments were run using a range of oscillation frequencies. It was found that transient eddy feedbacks significantly modified the response of the zonal winds to the external forcing and that this modification acted increasingly as a positive feedback as the period of the oscillation was increased. The eddy feedback was negative for periods less than 30 days. A simple linear positive feedback—that is, a negative drag—does not behave in this manner.

Robinson (1996) performed control simulations using the same two-level model and varying the strength of surface drag. The model was then modified by fixing the barotropic component of the zonal wind to the time average from the appropriate control run, the model was run with the barotropic zonal flow held fixed, and the eddy momentum fluxes were saved. Using these fluxes, the variability of the zonal flow that would have resulted if it had been permitted was calculated. For strong surface drag the zonal wind variations so diagnosed were far weaker, especially at low frequencies, than those generated in the control simulation. For weak surface drag, however, the diagnosed variations in the zonal flow were similar to those normally produced by the model. These experiments confirmed the presence of a positive eddy feedback in the model, operating primarily at low frequencies—periods of 30 or more days—and only in the presence of relatively strong surface drag.

In this paper we demonstrate that the above described features of the zonal index may be understood as straightforward consequences of the quasigeostrophic dynamics of the zonally averaged flow in the presence of surface drag and interior radiative damping, conditional on some plausible, but as yet unproven, assumptions about the behavior of transient baroclinic eddies:

• that baroclinic eddies are generated more vigorously in regions of stronger low-level baroclinicity, and

• that baroclinic eddies typically propagate away from their latitude of generation before they dissipate, and because of the sphericity of the earth, this propagation is more often equatorward than poleward.

The next section lays out the theoretical argument, and a few numerically calculated examples are shown in section 3. These results can be viewed as a tropospheric application of the principle of “downward control” (Haynes et al. 1991). Section 4 discusses the application of the results to understanding the zonal index.

2. Theory

a. Basic equations

Consider the equation for the evolution of the zonally averaged quasigeostrophic potential vorticity,

View Expanded

The potential vorticity in pressure coordinates is given by (Holton 1992)

View Expanded

where ζ is the relative vorticity, σ is a static stability parameter, and Φ is the geopotential. Taking the meridional derivative of (1) and using (2) and the geostrophic relation, we obtain an equation for the evolution of the zonally averaged zonal wind driven by the eddy fluxes of potential vorticity,

View Expanded

where the subscript t indicates differentiation with respect to time.

The boundary condition at the surface of the earth (or, more precisely, at the top of the planetary boundary layer, p = p_b) is derived from the zonally averaged thermodynamic equation. Defining, following Bretherton (1966), potential vorticity in a sheet at the lower boundary as

View Expanded

this boundary condition may be written,

View Expanded

Taking the meridional derivative yields the boundary condition at p = p_b for (3),

View Expanded

Later we make use of the fact derived by Bretherton (1966) that the globally integrated poleward eddy flux of potential vorticity vanishes, that is,

View Expanded

To close the problem it is necessary to specify the eddy fluxes of potential vorticity and the sources and sinks of potential vorticity. The sources and sinks result from the radiative damping of thermal perturbations and from boundary layer drag. Radiative damping is parameterized as a linear Newtonian damping of deviations of the zonally averaged temperature from the horizontally averaged temperature,

View Expanded

where τ_N is the radiative dissipation time. When this is included in (3), it becomes

View Expanded

Sources of potential vorticity at the boundary include the effects of radiative damping and vertical motion produced by Ekman pumping. Including both Ekman pumping and radiative damping in (6) yields the appropriate boundary condition for (8) at p = p_b,

View Expanded

where the Ekman dissipation time, τ_E, is typically about five days. At p = 0 the pressure vertical velocity vanishes. If it is assumed that the eddy heat flux also vanishes, the temperature cannot change, and

View Expanded

Because the zonally averaged problem is linear in the zonally averaged flow, these equations apply equally to the time-averaged zonal flow and to anomalies. The emphasis here is on understanding the zonal index, so that the latter interpretation is most relevant. We may, therefore, think of all the quantities in this paper—eddy fluxes and zonal mean responses—as deviations from some climatological mean.

b. The transient limit

We now consider two limiting cases of the system of equations (8), (9), and (10). The first is that of a transient response to the rearrangement of potential vorticity by eddies. If eddy forcing is turned on at some initial time, for times much shorter than the radiative and Ekman damping times, the response is given by the zonally symmetric potential vorticity inversion problem:

View Expanded

and at p = p_b,

View Expanded

A local maximum of northward eddy heat flux at the surface will cause a local maximum in the right-hand side of (12) and, therefore, a decrease in the westerly vertical shear. In other words, at the lower boundary, positive eddy heat fluxes cause a decrease in the baroclinicity of the zonal flow. It is expected, this will tend to suppress the generation of further baroclinic activity.

c. The steady limit

The other limiting case is the balance achieved after a long time between the eddy forcing and the dissipation. Given that the transient behavior invariably reduces the low-level baroclinicity, its enhancement at zero frequency is a necessary, if not sufficient, condition for the existence of positive feedback at low frequencies. In the steady limit (8) and (9) become

View Expanded

In (13) the eddy forcing is differentiated twice with respect to y, but there is no such differentiation on the left-hand side, so regions of strong eddy forcing are accompanied on their north and south flanks by lobes of oppositely signed responses in the zonal wind. From (14) it can be seen that poleward eddy heat fluxes at the surface need not be accompanied by decreased baroclinicity if there is a sufficiently strong westerly maximum with respect to latitude in the surface zonal wind. Then the increase with latitude in adiabatic cooling by the Ekman-induced vertical velocity can compensate for the poleward eddy heat flux.

d. Response to boundary forcing

A special case for the zonal mean equations is when the eddy forcing is only at the lower boundary. This case formally violates (7), but physically it corresponds to baroclinic eddy activity that is generated at the bottom boundary and propagates a great meridional distance from its source before dissipating. In the transient response to a poleward eddy heat flux at the lower boundary, baroclinicity is necessarily reduced, while the propagation of the eddies away from their source latitude accelerates the zonal winds at all levels. The steady response to lower-boundary forcing is much different. In the absence of interior forcing, (13) implies

View Expanded

The condition (10) at the top boundary requires that this constant vanish, and the response to surface forcing is, therefore, strictly barotropic.² For a barotropic solution, the lower-boundary condition (9) becomes

View Expanded

so that the barotropic response has the same meridional structure as the surface heat flux.

e. Forcing aloft

Because the steady response to surface forcing is barotropic, surface baroclinicity in the steady-state response is imposed by the distribution of the eddy flux of potential vorticity within the domain or, in other words, by where the eddy activity, once it is generated at the lower boundary, is absorbed. This can be seen by integrating (13) over the depth of the domain,

View Expanded

The surface baroclinicity in steady state is independent of the strength of the Ekman damping and of the vertical distribution of [υ*q*]. Enhanced low-level baroclinicity [negative left-hand side of (17)] in a region of baroclinic wave generation can, therefore, be maintained in steady state only if the bulk of the eddy activity propagates to neighboring latitudes before dissipating. Specifically, the generation region must coincide with the wings of the distribution of equatorward (negative) eddy flux of potential vorticity, where its meridional second derivative is negative.

3. Examples

For the zonal flow problem, the distribution of the eddy flux of potential vorticity is constrained only by (7). Reasonable choices can, however, be made by reference to the theory of baroclinic instability and to observations. In the time average there will be a region of positive [υ*q*] associated with the irreversible poleward transport of heat at the surface by baroclinic eddies. There must be compensating regions of equatorward fluxes of potential vorticity elsewhere in the atmosphere. In baroclinic life cycle calculations baroclinic eddies propagate to the upper troposphere and about 15° of latitude equatorward before depositing their wave activity [e.g., Edmon et al.’s (1980) diagnosis of the Simmons and Hoskins (1978) experiment]. Thus, we choose distributions of [υ*q*] that are positive in some region at the lower boundary and have a compensating negative region in the upper troposphere.

First, considering the response to eddy forcing confined to the boundary, we assume a Gaussian meridional distribution of the poleward eddy heat flux,

View Expanded

and we choose

View Expanded

with p_b = 1000 mb, and p_tr = 200 mb. Amplitudes here and below are chosen arbitrarily to give responses on the order of 10 in the appropriate units. From the results of Hartmann and Lo it appears that typical anomalous eddy forcing associated with the zonal index is on the order of 10 m s⁻¹ day⁻¹. Numerical solutions are obtained using successive overrelaxation on a 101 × 101 grid, extending meridionally from −7500 to 7500 km and from 1000 to 0 mb. The Coriolis parameter is set to a typical midlatitude value, f₀ = 10⁻⁴ s⁻¹. A constant lapse rate of 6 K km⁻¹ is used between 1000 and 200 mb, with a surface temperature of 288 K. The lapse rate is zero for pressures less than 200 mb. Second-order finite differences are used, except at the boundaries where one-sided differences are taken.

The transient response to this boundary forcing is shown in Fig. 1. There is an acceleration of the zonal flow, and decreasing low-level baroclinicity in the center of the domain is flanked by regions of increasing baroclinicity. The steady response to this same forcing (not shown) is, as described above, purely barotropic.

Examples that include interior fluxes are constructed by balancing the northward flux of potential vorticity in the boundary by a southward flux in the interior,

View Expanded

which together with (18) satisfies the condition that the globally integrated eddy flux of potential vorticity vanishes. Applying (17), the induced surface baroclinicity is positive for |y − y₀| > w/2, and the greatest induced surface baroclinicity is at |y − y₀| = (3/2)^1/2w. Note that increased baroclinicity occurs in two regions: the region of baroclinic eddy generation poleward of where the eddies are dissipated, and in a region equatorward of the dissipation region.

Figures 2a,b show the steady-state results with the parameters A_b = 10⁻⁵ m s⁻² × (p_b − p_tr) w = 1500 km, τ_E = 6.5 days, where p_l = 500 mb, p_tr = 200 mb, and the two values of the offset are y₀, = 0 and −1500 km. The superposition of the baroclinic response to the interior forcing, which shifts with the value of y₀, and the barotropic response to the surface forcing, centered at y = 0, is clearly evident.

Figure 3 shows the temporal adjustment of the surface baroclinicity at y = 0 for the same forcing parameters used to obtain Fig. 2b. The curves, obtained using different values of the dissipation parameters, have the characteristic shape of the difference between two decaying exponentials. The dependence on τ_E is shown in Fig. 3a. Consistent with (17), the asymptotic value of the baroclinicity at long times is independent of τ_E. The strength of the early reduction in baroclinicity and, therefore, the time required to approach the steady-state solution scale approximately linearly with τ_E. The radiative damping time (Fig. 3b) has less influence on the strongest negative baroclinicity. For larger values of τ_N, the approach to steady state is slower, but the final enhancement of the baroclinicity is greater.

4. Discussion

These results demonstrate that the mean flow modification by baroclinic eddies may indeed reinforce the low-level baroclinicity in their latitudes of generation, if the eddies propagate away before they dissipate. Figure 4 shows a schematic of this process. Baroclinic eddies propagate upward and equatorward, whereupon they break and are dissipated, resulting in a convergence of the Eliassen–Palm flux. This much of the picture is identical to that obtained for inviscid baroclinic life cycles (Edmon et al. 1980). In inviscid life cycle calculations, however, the thermal gradient is permanently expelled from the source latitudes of the baroclinic eddies. Here, the presence of surface friction allows a steady state to be achieved, wherein warming by the sinking branch of the transformed Eulerian mean circulation (Holton 1992, chapter 10) enhances the equatorward thermal gradient across the source latitudes.

If baroclinic eddies are generated more vigorously in regions of stronger low-level baroclinicity, this result provides a mechanism for the positive eddy feedback on zonal flow variations in midlatitudes. Furthermore, the present results are consistent with the features of zonal flow variations that were noted in the introduction.

1. Poleward drift of zonal flow anomalies. If, due to the sphericity of the earth, eddy activity predominantly propagates equatorward from its region of generation, then the low-level baroclinicity is reinforced poleward of where the eddies are generated. Thus, there is a poleward shift of the positive eddy feedback; and the anomalous low-level baroclinicity, vertically integrated zonal wind, and region of eddy generation will tend to drift poleward together. This poleward drift is limited, presumably, to latitudes where the climatological background baroclinicity is strong.

2. Meridional bandedness of the zonal wind and eddy forcing. In the steady state, the zonal flow response and the surface baroclinicity depend on the second meridional derivative of the eddy forcing. Thus, a single maximum in the westward eddy forcing gives rise to a deceleration of the zonal flow flanked by regions of acceleration, and a single maximum in the eddy forcing leads to multiple regions of strengthened baroclinicity.

3. Dependence of positive eddy feedback on surface drag. The steady response of the surface baroclinicity to a given eddy forcing is independent of the strength of the surface drag. In the presence of stronger surface drag, however, the approach to this steady state is accelerated, bringing the system more rapidly into a state of positive eddy feedback.

4. Frequency dependence of eddy feedback. Low-level baroclinicity in the region of eddy generation is reduced immediately after the eddy forcing is turned on but is increased in the steady state. The zonal flow response at lower frequencies will more closely resemble the steady-state response, while that at higher frequencies will resemble the transient behavior. Thus, we expect a negative eddy feedback at high frequencies and a positive feedback at low frequencies, which is consistent with what was found in a mechanistic model (Robinson 1994). This is confirmed by solutions obtained (not shown) for the temporal Fourier transforms of (8), (9), and (10).

Despite these promising elements of agreement with observations and models, the theory presented here is clearly incomplete in that the distribution of eddy forcing has simply been assumed, and only zonally symmetric dynamics have been considered. Furthermore, these results lead to positive eddy feedback only if it is true that baroclinic eddies are generated more vigorously at latitudes of stronger than usual low-level baroclinicity. In the absence of a complete theory that allows us to calculate eddy quantities in terms of the mean flow,³ the necessary next step is a careful analysis of the observed relationships between eddy fluxes and zonally averaged fields.